It is shown that there are primitive substitution filings with dense tile orientations invariant under n-fold rotation for n is an element of {2, 3, 4, 5, 6, 8). The proof for dense tile orientations uses a general result about irrationality of angles in certain parallelograms. (C) 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.